Accuracy of a coupled mixed and Galerkin finite element approximation for poroelasticity

Abstract

In this paper, we consider a combined mixed finite element and continuous Galerkin finite element formulation for a coupled flow and geomechanics model. We use the lowest order Raviart–Thomas finite elements for the spatial approximation of the flow variables and continuous piecewise linear finite elements for the deformation variable. This numerical approach appears to be a common choice in the existing reservoir engineering simulators. We focus on deriving error estimates in a discrete-in-time setting. Previous a priori error estimates described in the literature e.g. [2], [19], which are optimal, show first order convergence in space with respect to the $L^2$-norm for the pressure and for the average fluid velocity and also first order convergence in space with respect to the $H^1$-norm for the displacement. Here we prove one extra order of convergence for the displacement approximation with respect to the $L^2$-norm. We also demonstrate that, by including a post-processing step in the scheme, the order of convergence of the approximation of pressure can be improved. Even though this result is critical for deriving the $L^2$-norm error estimates for the approximation of the deformation variable, surprisingly the corresponding gain of one convergence order holds independently of including or not the post-processing step in the method.